Asymptotically flat three-manifolds contain minimal planes

Seiberg-Witten Equations on Asymptotically Flat Three Manifolds

We construct the Seiberg-Witten theory on asymptotically flat three manifolds and describe the structure of the moduli space. The analysis should serve as the basis for many applications in 3-manifold topology, including a proof of the equivalence of the Seiberg-Witten invariant of 3-manifolds and the Reidemeister torsion [7].

Curvature Estimates in Asymptotically Flat Lorentzian Manifolds

We consider an asymptotically flat Lorentzian manifold of dimension (1, 3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when the ADM energy tends to zero.

A Note on Existence and Non-existence of Minimal Surfaces in Some Asymptotically Flat 3-manifolds

Motivated by problems on apparent horizons in general relativity, we prove the following theorem on minimal surfaces: Let g be a metric on the three-sphere S satisfying Ric(g) ≥ 2g. If the volume of (S, g) is no less than one half of the volume of the standard unit sphere, then there are no closed minimal surfaces in the asymptotically flat manifold (S \ {P}, Gg). Here G is the Green’s function...

Inequalities for the ADM-mass and capacity of asymptotically flat manifolds with minimal boundary

We present some recent developments involving inequalities for the ADM-mass and capacity of asymptotically flat manifolds with boundary. New, more general proofs of classic Euclidean estimates are also included. The inequalities are rigid and valid in all dimensions, and constitute a step towards proving the Riemannian Penrose inequality in arbitrary dimensions.

Area-minimizing Projective Planes in Three-manifolds